Typical Theist Argument: "The odds of the universe being the way it is by chance are only 1: [really huge number], ergo, it must have been created by a deity!"
Atheism Resource:
Allow me to refute this argument by way of demonstration: Suppose that I took a deck of cards, shuffled it well, and then spread the cards out, face up on the table. In fact, I will do just that, and post a picture of the outcome so we can both be on the same page.
Take a look at this deck of 54 cards (52 plus the Jokers). Are you particularly awestruck by the ordering of those cards? My guess is that you probably are not. But do you know what the odds of getting all 54 cards in that exact order are? 1: 2*10^71.
To write it out longhand, the odds are 1:230,843,697,339,241,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
So why did you not find anything impressive in the ordering of these cards despite the nigh impossible odds? The answer is, quite simply, that each and every one of those 2*10^71 is equally probable, so it should cause no surprise that one of them turned up.
Now let's turn the clock back and do the same thing with just a minor tweak: Let's suppose that I shuffled the deck of cards and then I predicted the exact order of all 54 cards in that deck. Suppose that we then we spread the cards out on the table and found that my prediction was exactly correct. Under those circumstances, I expect that you would be (appropriately) awestruck about the ordering of the cards.
So what is different about these two situations? Once again, the answer is quite simple: In the first scenario, we were not seeking to get any particular ordering of the cards. Each possible permutation was as likely as any other, and there was no one permutation that we were actively seeking to achieve. In the second scenario, however, my act of predicting created a targeted outcome. By predicting the order of all 54 cards, I took one of those 2*10^71 possible outcomes, and established it as the one and only outcome that I was attempting to achieve.
In the same way, saying "the odds of the universe being the way that it is are minuscule" is only impressive if one accepts the premise that there was a being who pre-existed the creation of the universe, and who specifically intended that the universe should be as it is, as opposed to any of the near-infinite number of other possible ways that it could have been, which is exactly the point in want of proving!
If one does not assume the existence of a deity as a premise (which one cannot, if one intends for the argument to prove the existence of a deity), one must accept that every possible way that the universe could be is equally likely, and that the universe being the way that it is is no more unlikely than any other possible way that it could have been.
TL;DR: Looking at the odds of the universe being as it is, and concluding that a deity must exist is simply a matter of "begging the question" (taking the conclusion that one intends to prove, and including it as a premise of ones argument).